SRB-like Measures for $C^0$ Dynamics

Eleonora Catsigeras¹ ; Heber Enrich¹

¹Instituto de Matemática y Estadística Prof. Ing. Rafael Laguardia Facultad de Ingeniería Universidad de la República Av. Herrera y Reissig 565 C.P.11300, Montevideo, Uruguay

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011) no. 2, pp. 151-164

Résumé

For any continuous map $f\colon M \to M$ on a compact manifold $M$, we define SRB-like (or observable) probabilities as a generalization of Sinai–Ruelle–Bowen (i.e. physical) measures. We prove that $f$ always has observable measures, even if SRB measures do not exist. We prove that the definition of observability is optimal, provided that the purpose of the researcher is to describe the asymptotic statistics for Lebesgue almost all initial states. Precisely, the never empty set ${\mathcal O}$ of all observable measures is the minimal weak$^*$ compact set of Borel probabilities in $M$ that contains the limits (in the weak$^*$ topology) of all convergent subsequences of the empirical probabilities $ \{(1/n) \sum_{j= 0}^{n-1}\delta_{f^j(x)}\}_{n \geq 1}$, for Lebesgue almost all $x \in M$. We prove that any isolated measure in ${\mathcal O}$ is SRB. Finally we conclude that if ${\mathcal O}$ is finite or countably infinite, then there exist (countably many) SRB measures such that the union of their basins covers $M$ Lebesgue a.e.

Détail

DOI : 10.4064/ba59-2-5

Keywords: continuous map colon compact manifold define srb like observable probabilities generalization sinai ruelle bowen physical measures prove always has observable measures even srb measures exist prove definition observability optimal provided purpose researcher describe asymptotic statistics lebesgue almost initial states precisely never empty set mathcal observable measures minimal weak * compact set borel probabilities contains limits weak * topology convergent subsequences empirical probabilities sum n delta geq lebesgue almost prove isolated measure mathcal srb finally conclude mathcal finite countably infinite there exist countably many srb measures union their basins covers lebesgue nbsp